A method for using a numerical method to design reference signals for multiple input multiple output (MIMO) systems is described. An input multiple input multiple output signal is determined. A nearest tight frame to one or more given structured vectors is obtained. One or more structured vectors from the nearest tight frame are obtained. Orthogonal subsets are computed for a plurality of sequences, wherein each of the subsets is replaced with a matrix that comprises a function of one or more elements in each of the plurality of sequences and an identity matrix associated with the one or more elements. The one or more structured vectors are projected onto the space of circulant matrices. One or more classes of matrices associated with reference signals are outputted.